Differential Mueller matrix of a depolarizing homogeneous medium and its relation to the Mueller matrix logarithm

The different z-dependence and non-commutativity of the two components of the differential Mueller matrix of a homogeneous depolarizing medium prevent its formal identification with the Mueller matrix logarithm. By using a classic linear differential equation expansion, we advance a procedure for the extraction of the elementary polarization properties, in terms of mean values and variances-covariances, from the Mueller matrix logarithm. The approximate solution, based on the immediate identification of the differential matrix with the matrix logarithm, turns out to remain satisfactory up to relatively high depolarization levels. Physically interpreted experimental examples from the literature illustrate the formal developments.